Theorem dihglbcN

Description: Isomorphism H of a lattice glb when the glb is not under the fiducial hyperplane W . (Contributed by NM, 26-Mar-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dihglbc.b	$$⊢B={\mathrm{Base}}_K$$
	dihglbc.g	$$⊢G=\mathrm{glb}\left(K\right)$$
	dihglbc.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	dihglbc.i	$$⊢I=\mathrm{DIsoH}\left(K\right)\left(W\right)$$
	dihglbc.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
Assertion	dihglbcN	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(S\subseteq B\wedge S\ne \varnothing \right)\wedge \neg G\left(S\right)\underset{\dot{}}{\le }W\right)\to I\left(G\left(S\right)\right)=\bigcap _{x\in S}I\left(x\right)$$

Proof

Step	Hyp	Ref	Expression
1		dihglbc.b	$$⊢B={\mathrm{Base}}_K$$
2		dihglbc.g	$$⊢G=\mathrm{glb}\left(K\right)$$
3		dihglbc.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
4		dihglbc.i	$$⊢I=\mathrm{DIsoH}\left(K\right)\left(W\right)$$
5		dihglbc.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
6		eqid	$$⊢\mathrm{join}\left(K\right)=\mathrm{join}\left(K\right)$$
7		eqid	$$⊢\mathrm{meet}\left(K\right)=\mathrm{meet}\left(K\right)$$
8		eqid	$$⊢\mathrm{Atoms}\left(K\right)=\mathrm{Atoms}\left(K\right)$$
9		eqid	$$⊢\mathrm{oc}\left(K\right)\left(W\right)=\mathrm{oc}\left(K\right)\left(W\right)$$
10		eqid	$$⊢\mathrm{LTrn}\left(K\right)\left(W\right)=\mathrm{LTrn}\left(K\right)\left(W\right)$$
11		eqid	$$⊢\mathrm{trL}\left(K\right)\left(W\right)=\mathrm{trL}\left(K\right)\left(W\right)$$
12		eqid	$$⊢\mathrm{TEndo}\left(K\right)\left(W\right)=\mathrm{TEndo}\left(K\right)\left(W\right)$$
13		eqid	$$⊢\left(\iota g\in \mathrm{LTrn}\left(K\right)\left(W\right)|g\left(\mathrm{oc}\left(K\right)\left(W\right)\right)=q\right)=\left(\iota g\in \mathrm{LTrn}\left(K\right)\left(W\right)|g\left(\mathrm{oc}\left(K\right)\left(W\right)\right)=q\right)$$
14	1 2 3 4 5 6 7 8 9 10 11 12 13	dihglbcpreN	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(S\subseteq B\wedge S\ne \varnothing \right)\wedge \neg G\left(S\right)\underset{\dot{}}{\le }W\right)\to I\left(G\left(S\right)\right)=\bigcap _{x\in S}I\left(x\right)$$